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A052841 E.g.f.: 1/(exp(x)*(2-exp(x))). 10
1, 0, 2, 6, 38, 270, 2342, 23646, 272918, 3543630, 51123782, 811316286, 14045783798, 263429174190, 5320671485222, 115141595488926, 2657827340990678, 65185383514567950, 1692767331628422662, 46400793659664205566, 1338843898122192101558, 40562412499252036940910 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

From Michael Somos, Mar 04 2004: (Start)

Stirling transform of A005359(n)=[0,2,0,24,0,720,...] is a(n)=[0,2,6,38,270,...].

Stirling transform of -(-1)^n*A052657(n-1)=[0,0,2,-6,48,-240,...] is a(n-1)=[0,0,2,6,38,270,...].

Stirling transform of -(-1)^n*A052558(n-1)=[1,-1,4,-12,72,-360,...] is a(n-1)=[1,0,2,6,38,270,...].

Stirling transform of 2*A052591(n)=[2,4,24,96,...] is a(n+1)=[2,6,38,270,...].

(End)

Also the central moments of a Geometric(1/2) random variable (for example the number of coin tosses until the first head). - Svante Janson, Dec 10 2012

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

C. G. Bower, Transforms (2)

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 808

Svante Janson, Euler-Frobenius numbers and rounding, preprint arXiv:1305.3512 [math.PR], 2013.

FORMULA

O.g.f.: Sum_{n>=0} (2*n)! * x^(2*n) / Product_{k=1..2*n} (1-k*x). - Paul D. Hanna, Jul 20 2011

a(n) = (A000670(n) + (-1)^n)/2 = Sum_{k>=0} (k-1)^n/2^(k+1). - Vladeta Jovovic, Feb 02 2003

Also, a(n) = Sum[k=0..[n/2], (2k)!*Stirling2(n, 2k)]. - Ralf Stephan, May 23 2004

a(n) = D^n(1/(1-x^2)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000670 and A005649. - Peter Bala, Nov 25 2011

E.g.f.: 1/2/G(0) where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k*(k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 22 2012

a(n) ~ n!/(4*(log(2))^(n+1)). - Vaclav Kotesovec, Aug 10 2013

a(n) = (h(n)+(-1)^n)/2 where h(n) = Sum_{k=0..n} E(n,k)*2^k and E(n,k) the Eulerian numbers A173018 (see also A156365). - Peter Luschny, Sep 19 2015

MAPLE

spec := [S, {B=Prod(C, C), C=Set(Z, 1 <= card), S=Sequence(B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

P := proc(n, x) option remember; if n = 0 then 1 else

(n*x+2*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x); expand(%) fi end:

A052841 := n -> subs(x=2, P(n, x)):

seq(A052841(n), n=0..21); # Peter Luschny, Mar 07 2014

h := n -> add(combinat:-eulerian1(n, k)*2^k, k=0..n):

a := n -> (h(n)+(-1)^n)/2: seq(a(n), n=0..21); # Peter Luschny, Sep 19 2015

MATHEMATICA

a[n_] := If[n == 0, 1, (PolyLog[-n, 1/2]/2 + (-1)^n)/2]; (* or *)

a[n_] := HurwitzLerchPhi[1/2, -n, -1]/2; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 19 2016, after Vladeta Jovovic *)

PROG

(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst(1/(1-y^2), y, exp(x+x*O(x^n))-1), n))

(PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!*x^(2*m)/prod(k=1, 2*m, 1-k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */

CROSSREFS

Inverse binomial transform of A000670.

Cf. A052558, A052591, A052657, A005359, A173018, A156365.

Sequence in context: A027322 A085447 A078673 * A275557 A197972 A068184

Adjacent sequences:  A052838 A052839 A052840 * A052842 A052843 A052844

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

Edited by N. J. A. Sloane, Sep 06 2013

STATUS

approved

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Last modified December 10 04:09 EST 2016. Contains 278993 sequences.